Question

Starting with the general formulas for gravitational force and centripetal force, derive the relationship between speed (v) and orbital radius (r) for a circular orbit. Show your work. Is the relationship consistent with what you observed in Step 8? Explain.The observation in step 8 was: The velocity needed to increase to obtain a circular orbit at the smaller radius.

Answer 1

Consider:

The mass of the planet is M.

The mass of the orbiting body is m.

The radius of the orbital in which the body is revolving around the planet is r.

The speed of the orbiting body is v.

Solution:

The gravitational force acting on the orbiting body due to the planet is,

$F=\frac{\text{GMm}}{r^2}$

where G is the gravitational constant,

This gravitational force acting on the orbiting body helps the body to move in a circular motion around the planet.

The centripetal force acting on the orbiting body is,

$F=\frac{mv^2}{r}$

As the centripetal force and the gravitational force acting on the orbiting body are the same, thus,

$\begin{gathered} \frac{\text{GMm}}{r^2}=\frac{mv^2}{r} \\ \frac{GM}{r}=v^2 \\ v=\sqrt[]{\frac{GM}{r}} \end{gathered}$

From the formula, the speed of the orbiting body is inversely proportional to the orbital radius.

Thus, with the decrease in the orbital radius of the orbiting body, the velocity of the body increases.

Hence, the relation between the velocity and orbital radius is consistent with the observation in step 8.